This chapter focuses on the observational side of astronomy &mdash; what we see when we look at the sky.

This chapter focuses on the observational side of astronomy &mdash; what we see when we look at the sky.

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====Catalogues====

====Catalogues====

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As described in section [sec:astronomicalcatalogues], various star catalogues assign numbers to stars, which are often used in addition to other names. Stellarium gets it's star data from the Hipparcos catalogue, and as such stars in Stellarium are generally referred to with their Hipparcos number, e.g. “HP 62223”. Figure [fig:starnames] shows the information Stellarium displays when a star is selected. At the top, the common name and Flamsteed designation are shown, followed by the RA/Dec coordinates, apparent magnitude, distance and Hipparcos number.

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As described in section [[Star_Catalogue|star catalogue]], various star catalogues assign numbers to stars, which are often used in addition to other names. Stellarium gets it's star data from the Hipparcos catalogue, and as such stars in Stellarium are generally referred to with their Hipparcos number, e.g. “HP 62223”. Figure [fig:starnames] shows the information Stellarium displays when a star is selected. At the top, the common name and Flamsteed designation are shown, followed by the RA/Dec coordinates, apparent magnitude, distance and Hipparcos number.

http://stellarium.org/wikiimg/UserManual/names.png

http://stellarium.org/wikiimg/UserManual/names.png

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By splitting the light from a star using a prism attached to a telescope and measuring the relative intensities of the colours of light the star emits &mdash; the ''spectra'' &mdash; a great deal of interesting information can be discovered about a star including its surface temperature, and the presence of various elements in its atmosphere.

By splitting the light from a star using a prism attached to a telescope and measuring the relative intensities of the colours of light the star emits &mdash; the ''spectra'' &mdash; a great deal of interesting information can be discovered about a star including its surface temperature, and the presence of various elements in its atmosphere.

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{| class="standard"

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{| class="guidetable"

|+ Spectral Types

|+ Spectral Types

| Spectral Type || Surface Temperature (&deg;K) || Star Colour

| Spectral Type || Surface Temperature (&deg;K) || Star Colour

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A ''luminosity class'' is an indication of the type of star &mdash; whether it is main sequence, a giant or a dwarf. Luminosity classes are denoted by a number in roman numerals, as described in table [fig:luminosityclass].

A ''luminosity class'' is an indication of the type of star &mdash; whether it is main sequence, a giant or a dwarf. Luminosity classes are denoted by a number in roman numerals, as described in table [fig:luminosityclass].

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{| class="standard"

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{| class="guidetable"

|+ Luminosity Class

|+ Luminosity Class

| ''Luminosity class'' || ''Description''

| ''Luminosity class'' || ''Description''

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Using you hand in this way is not very precise, but it's close enough to give you some way to translate an idea like “Mars will be 45&deg; above the Southeastern horizon at 21:30”. Of course, there is variation from person to person, but the variation is compensated for somewhat by the fact that people with long arms tend to have larger hands. In exercise [exercise:handyangles], you will work out your own “handy angles”.

Using you hand in this way is not very precise, but it's close enough to give you some way to translate an idea like “Mars will be 45&deg; above the Southeastern horizon at 21:30”. Of course, there is variation from person to person, but the variation is compensated for somewhat by the fact that people with long arms tend to have larger hands. In exercise [exercise:handyangles], you will work out your own “handy angles”.

The Sun

Without a doubt, the most prominent object in the sky is the Sun. The Sun is so bright that when it is in the sky, it's light is scattered by the atmosphere to such an extent that almost all other objects in the sky are rendered invisible.

The Sun is a star like many others but it is much closer to the Earth at approximately 150 million kilometres. The next nearest star, Proxima Centuri is approximately 260,000 times further away from us than the Sun! The Sun is also known as Sol, it's Latin name.

Over the course of a year, the Sun appears to move round the celestial sphere in a great circle known as the ecliptic. Stellarium can draw the ecliptic on the sky. To toggle drawing of the ecliptic, press the 4 or , key.

WARNING: Looking at the Sun can permanently damage the eye. Never look at the Sun without using the proper filters! By far the safest way to observe the Sun it to look at it on a computer screen, courtesy of Stellarium!

Stars

The Sun is just one of billions of stars. Even though many stars have a much greater absolute magnitude than the Sun (the give out more light), they have an enormously smaller apparent magnitude due to their large distance. Stars have a variety of forms — different sizes, brightnesses, temperatures, and colours. Measuring the position, distance and attributes of the stars is known as astrometry, and is a major part of observational astronomy.

Multiple Star Systems

Many stars have a stellar companions. As many as six stars can be found orbiting one-another in close association. Such associations are known a multiple star systems — binary systems being the most common with two stars. Multiple star systems are more common than solitary stars, putting our Sun in the minority group.

Sometimes multiple stars orbit one-another in a way that means one will periodically eclipse the other. These eclipsing binaries or Algol variables.

Optical Doubles & Optical Multiples

Sometimes two or more stars appear to be very close to one another in the sky, but in fact have great separation, being aligned from the point of view of the observer but of different distances. Such pairings are known as optical doubles and optical multiples.

Constellations

The constellations are groupings of stars that are visually close to one another in the sky. The actual groupings are fairly arbitrary — different cultures have group stars together into different constellations. In many cultures, the various constellations have been associated with mythological entities. As such people have often projected pictures into the skies as can be seen in figure [fig:ursamajor] which shows the constellation of Ursa Major. On the left is a picture with the image of the mythical Great Bear, on the right only a line-art version is shown. The seven bright stars of Ursa Major are widely recognised, known variously as “the plough”, the “pan-handle”, and the “big dipper”. This sub-grouping is known as an asterism — a distinct grouping of stars. On the right, the picture of the bear has been removed and only a constellation diagram remains.

Stellarium can draw both constellation diagrams and artistic representations of the constellations. Multiple sky cultures are supported: Western, Polynesian, Egyptian and Chinese constellations are available, although at time of writing the non-Western constellations are not complete, and as yet there are no artistic representations of these sky-cultures.

Aside from historical and mythological value, to the modern astronomer the constellations provide a way to segment the sky for the purposes of describing locations of objects, indeed one of the first tasks for an amateur observer is learning the constellations — the process of becoming familiar with the relative positions of the constellations, at what time of year a constellation is visible, and in which constellations observationally interesting objects reside. Internationally, astronomers have adopted the Western (Greek/Roman) constellations as a common system for segmenting the sky. As such some formalisation has been adopted, each constellation having a proper name, which is in Latin, and a three letter abbreviation of that name. For example, Ursa Major has the abbreviation UMa.

Star Names

Stars can have many names. The brighter stars often have common names relating to mythical characters from the various traditions. For example the brightest star in the sky, Sirius is also known as The Dog Star (the name Canis Major — the constellation Sirius is found in — is Latin for “The Great Dog”).

There are several more formal naming conventions that are in common use.

Bayer Designation

German astronomer Johan Bayer devised one such system in the 16-17th century. His scheme names the stars according to the constellation in which they lie prefixed by a lower case Greek letter, starting at α for the brightest star in the constellation and proceeding with β, γ, ... in descending order of apparent magnitude. For example, such a Bayer Designation for Sirius is “α Canis Majoris” (note that the genitive form of the constellation name is used). There are some exceptions to the descending magnitude ordering, and some multiple stars (both real and optical) are named with a numerical superscript after the Greek letter, e.g. π1... π6 Orionis.

Flamsteed Designation

English astronomer John Flamsteed numbered stars in each constellation in order of increasing right ascension followed by the form of the constellation name, for example “61 Cygni”.

Catalogues

As described in section star catalogue, various star catalogues assign numbers to stars, which are often used in addition to other names. Stellarium gets it's star data from the Hipparcos catalogue, and as such stars in Stellarium are generally referred to with their Hipparcos number, e.g. “HP 62223”. Figure [fig:starnames] shows the information Stellarium displays when a star is selected. At the top, the common name and Flamsteed designation are shown, followed by the RA/Dec coordinates, apparent magnitude, distance and Hipparcos number.

Spectral Type & Luminosity Class

Stars have many different colours. Seen with the naked eye most appear to be white, but this is due to the response of the eye — at low light levels the eye is not sensitive to colour. Typically the unaided eye can start to see differences in colour only for stars that have apparent magnitude brighter than 1. Betelgeuse, for example has a distinctly red tinge to it, and Sirius appears to be blue.

By splitting the light from a star using a prism attached to a telescope and measuring the relative intensities of the colours of light the star emits — the spectra — a great deal of interesting information can be discovered about a star including its surface temperature, and the presence of various elements in its atmosphere.

Spectral Types

Spectral Type

Surface Temperature (°K)

Star Colour

O

28,000—50,000

Blue

B

10,000—28,000

Blue-white

A

7,500—10,000

White-blue

F

6,000—7,500

Yellow-white

G

4,900—6,000

Yellow

K

3,500—4,900

Orange

M

2,000—3,500

Red

Astronomers groups stars with similar spectra into spectral types, denoted by one of the following letters: O, B, A, F, G, K and M. Type O stars have a high surface temperature (up to around 50,000°K) while the at other end of the scale, the M stars are red and have a much cooler surface temperature, typically 3000°K. The Sun is a type G star with a surface temperature of around 5,500°K. Spectral types may be further sub-divided using a numerical suffixes ranging from 0-9 where 0 is the hottest and 9 is the coolest. Table [fig:spectraltype] shows the details of the various spectral types.

For about 90% of stars, the absolute magnitude increases as the spectral type tends to the O (hot) end of the scale. Thus the whiter, hotter stars tend to have a greater luminosity. These stars are called main sequence stars. There are however a number of stars that have spectral type at the M end of the scale, and yet they have a high absolute magnitude. These stars have a very large size, and consequently are known as giants, the largest of these known as super-giants.

There are also stars whose absolute magnitude is very low regardless of the spectral class. These are known as dwarf stars, among them white dwarfs and brown dwarfs.

A luminosity class is an indication of the type of star — whether it is main sequence, a giant or a dwarf. Luminosity classes are denoted by a number in roman numerals, as described in table [fig:luminosityclass].

Luminosity Class

Luminosity class

Description

Ia, Ib

Super-giants

II

Bright giants

III

Normal giants

IV

Sub-giants

V

Main sequence

VI

Sub-dwarfs

VII

White-dwarfs

Plotting the luminosity of stars against their spectral type/surface temperature, gives a diagram called a Hertzsprung-Russell diagram (after the two astronomers Ejnar Hertzsprung and Henry Norris Russell who devised it). A slight variation of this is see in figure [fig:colourmag] (which is technically a colour/magnitude plot).

Variables

Most stars are of nearly constant luminosity. The Sun is a good example of one which goes through relatively little variation in brightness (usually about 0.1% over an 11 year solar cycle). Many stars, however, undergo significant variations in luminosity, and these are known as variable stars. There are many types of variable stars falling into two categories intrinsic and extrinsic.

Intrinsic variables are stars which have intrinsic variations in brightness, that is the star itself gets brighter and dimmer. There are several types of intrinsic variables, probably the best-known and more important of which is the Cepheid variable whose luminosity is related to the period with which it's brightness varies. Since the luminosity (and therefore absolute magnitude) can be calculated, Cepheid variables may be used to determine the distance of the star when the annual parallax is too small to be a reliable guide.

Extrinsic variables are stars of constant brightness that show changes in brightness as seen from the Earth. These include rotating variables, or stars whose apparent brightness change due to rotation, and eclipsing binaries.

Our Moon

The Moon is the large satellite which orbits the Earth approximately every 28 days. It is seen as a large bright disc in the early night sky that rises later each day and changes shape into a crescent until it disappears near the Sun. After this it rises during the day then gets larger until it again becomes a large bright disc again.

Phases of the Moon

As the moon moves round its orbit, the amount that is illuminated by the sun as seen from a vantage point on Earth changes. The result of this is that approximately once per orbit, the moon's face gradually changes from being totally in shadow to being fully illuminated and back to being in shadow again. This process is divided up into various phases as described in table [tab:moonphases].

The Major Planets

Unlike the stars whose relative positions remain more or less constant, the planets seem to move across the sky over time (the word “planet” comes from the Greek for “wanderer”). The planets are, like the Earth, massive bodies that are in orbit around the Sun. Until 2006 there was no formal definition of a planet leading to some confusion about the classification for some bodies widely regarded as being planets, but which didn't seem to fit with the others.

In 2006 the International Astronomical Union defined a planet as a celestial body that, within the Solar System:

is in orbit around the Sun

has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape; and

has cleared the neighbourhood around its orbit

or within another system:

is in orbit around a star or stellar remnants

has a mass below the limiting mass for thermonuclear fusion of deuterium; and

is above the minimum mass/size requirement for planetary status in the Solar System.

Moving from the Sun outwards, the major planets are: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune. Since the formal definition of a planet in 2006 Pluto has been relegated to having the status of dwarf planet along with bodies such as Ceres and Eris. See figure

Terrestrial Planets

The planets closest to the sun are called collectively the terrestrial planets. The terrestrial planets are: Mercury, Venus, Earth and Mars.

The terrestrial planets are relatively small, comparatively dense, and have solid rocky surface. Most of their mass is made from solid matter, which is mostly rocky and/or metallic in nature.

Jovian Planets

Jupiter, Saturn, Uranus and Neptune make up the Jovian planets. They are much more massive than the terrestrial planets, and do not have a solid surface. Jupiter is the largest of all the planets with a mass over 300 times that of the Earth!

The Jovian planets do not have a solid surface - the vast majority of their mass being in gaseous form (although they may have rocky or metallic cores). Because of this, they have an average density which is much less than the terrestrial planets. Saturn's mean density is only about 0.7 g/cm3 - it would float in water!

The Minor Planets

As well as the Major Planets, the solar system also contains innumerable smaller bodies in orbit around the Sun. These are generally classed as the minor planets, or planetoids, and include asteroids, and [sometimes?] comets.

Asteroids

Asteroids are celestial bodies orbiting the Sun in more or less regular orbits mostly between Mars and Jupiter. They are generally rocky bodies like the inner (terrestrial) planets, but of much smaller size. There are countless in number ranging in size from about ten meters to thousands of kilometres.

Comets

A comet is a small body in the solar system that orbits the Sun and (at least occasionally) exhibits a coma (or atmosphere) and/or a tail.

Comets have a very eccentric orbit (very elliptical), and as such spend most of their time a very long way from the Sun. Comets are composed of rock, dust and ices. When they come close to the Sun, the heat evaporates the ices, causing a gaseous release. This gas, and loose material which comes away from the body of the comet is swept away from the Sun by the Solar wind, forming the tail.

Comets whose orbit brings them close to the Sun more frequently than every 200 years are considered to be short period comets, the most famous of which is probably Comet Halley, named after the British astronomer Edmund Halley, which has an orbital period of roughly 76 years.

Galaxies

Stars, it seems, are gregarious - they like to live together in groups. These groups are called galaxies. The number of stars in a typical galaxy is literally astronomical - many billions - sometimes ever hundreds of billions of stars!

Our own star, the sun, is part of a galaxy. When we look up at the night sky, all the stars we can see are in the same galaxy. We call our own galaxy the Milky Way (or sometimes simply “the Galaxy”).

Other galaxies appear in the sky as dim fuzzy blobs. Only four are normally visible to the naked eye. The Andromeda galaxy (M31) visible in the Northern hemisphere, the two Magellanic clouds, visible in the Southern hemisphere, and the home galaxy Milky Way, visible in parts from north and south under dark skies.

There are thought to be billions of galaxies in the universe comprised of an unimaginably large number of stars.

The vast majority of galaxies are so far away that they are very dim, and cannot be seen without large telescopes, but there are dozens of galaxies which may be observed in medium to large sized amateur instruments. Stellarium includes images of many galaxies, including the Andromeda galaxy (M31), the Pinwheel Galaxy (M101), the Sombrero Galaxy (M104) and many others.

Astronomers classify galaxies according to their appearance. Some classifications include spiral galaxies, elliptical galaxies, lenticular galaxies and irregular galaxies.

The Milky Way

It's a little hard to work out what our galaxy would look like from far away, because when we look up at the night sky, we are seeing it from the inside. All the stars we can see are part of the Milky Way, and we can see them in every direction. However, there is some structure. There is a higher density of stars in particular places.

There is a band of very dense stars running right round the sky in huge irregular stripe. Most of these stars are very dim, but the overall effect is that on very dark clear nights we can see a large, beautiful area of diffuse light in the sky. It is this for which we name our galaxy.

The reason for this effect is that our galaxy is somewhat like a disc, and we are off to one side. Thus when we look towards the centre of the disc, we see more a great concentration of stars (there are more star in that direction). As we look out away from the centre of the disc we see fewer stars - we are staring out into the void between galaxies!

Nebulae

Seen with the naked eye, binoculars or a small telescope, a nebula (plural nebulae) are fuzzy patches on the sky. Historically, the term referred to any extended object, but the modern definition excludes some types of object such as galaxies.

Observationally, nebulae are popular objects for amateur astronomers - they exhibit complex structure, spectacular colours and a wide variety of forms. Many nebulae are bright enough to be seen using good binoculars or small to medium sized telescopes, and are a very photogenic subject for astro-photographers.

Nebulae are associated with a variety of phenomena, some being clouds of interstellar dust and gas in the process of collapsing under gravity, some being envelopes of gas thrown off during a supernova event (so called supernova remnants), yet others being the remnants of solar systems around dead stars (planetary nebulae).

Examples of nebulae for which Stellarium has images include the Crab Nebula (M1), which is a supernova remnant and the Dumbbell Nebula (M27) which is a planetary nebula.

Meteoroids

These objects are small pieces of space debris left over from the early days of the solar system that orbit the Sun. They come in a variety of shapes, sizes an compositions, ranging from microscopic dust particles up to about ten meters across.

Sometimes these objects collide with the Earth. The closing speed of these collisions is generally extremely high (tens or kilometres per second). When such an object ploughs through the Earth's atmosphere, a large amount of kinetic energy is converted into heat and light, and a visible flash or streak can often be seen with the naked eye. Even the smallest particles can cause these events which are commonly known as shooting stars.

While smaller objects tend to burn up in the atmosphere, larger, denser objects can penetrate the atmosphere and strike the surface of the planet, sometimes leaving meteor craters.

Sometimes the angle of the collision means that larger objects pass through the atmosphere but do not strike the Earth. When this happens, spectacular fireballs are sometimes seen.

Meteoroids is the name given to such objects when they are floating in space.

A Meteor is the name given to the visible atmospheric phenomenon.

Meteorites is the name given to objects that penetrate the atmosphere and land on the surface.

Eclipses

Eclipses occur when an apparently large celestial body (planet, moon etc.) moves between the observer (that's you!) and a more distant object - the more distant object being eclipsed by the nearer one.

Solar Eclipses

Solar eclipses occur when our Moon moves between the Earth and the Sun. This happens when the inclined orbit of the Moon causes its path to cross our line of sight to the Sun. In essence it is the observer falling under the shadow of the moon.

There are three types of solar eclipses:

Partial The Moon only covers part of the Sun's surface.

Total The Moon completely obscures the Sun's surface.

Annular The Moon is at aphelion (furthest from Earth in its elliptic orbit) and its disc is too small to completely cover the Sun. In this case most of the Sun's disc is obscured - all except a thin ring around the edge.

Lunar Eclipses

Lunar eclipses occur when the Earth moves between the Sun and the Moon, and the Moon is in the Earth's shadow. They occur under the same basic conditions as the solar eclipse but can occur more often because the Earth's shadow is so much larger than the Moon's.

Total lunar eclipses are more noticeable than partial eclipses because the Moon moves fully into the Earth's shadow and there is very noticeable darkening. However, the Earth's atmosphere refracts light (bends it) in such a way that some sunlight can still fall on the Moon's surface even during total eclipses. In this case there is often a marked reddening of the light as it passes through the atmosphere, and this can make the Moon appear a deep red colour.

Catalogues

Astronomers have made various catalogues of objects in the heavens. Stellarium makes use of several well known astronomical catalogues.

Hipparcos

Hipparcos (for High Precision Parallax Collecting Satellite) was an astrometry mission of the European Space Agency (ESA) dedicated to the measurement of stellar parallax and the proper motions of stars. The project was named in honour of the Greek astronomer Hipparchus.

Ideas for such a mission dated from 1967, with the mission accepted by ESA in 1980. The satellite was launched by an Ariane 4 on 8 August 1989. The original goal was to place the satellite in a geostationary orbit above the earth, however a booster rocket failure resulted in a highly elliptical orbit from 315 to 22,300 miles altitude. Despite this difficulty, all of the scientific goals were accomplished. Communications were terminated on 15 August 1993.

The program was divided in two parts: the Hipparcos experiment whose goal was to measure the five astrometric parameters of some 120,000 stars to a precision of some 2 to 4 milli arc-seconds and the Tycho experiment, whose goal was the measurement of the astrometric and two-colour photometric properties of some 400,000 additional stars to a somewhat lower precision.

The final Hipparcos Catalogue (120,000 stars with 1 milli arc-second level astrometry) and the final Tycho Catalogue (more than one million stars with 20-30 milli arc-second astrometry and two-colour photometry) were completed in August 1996. The catalogues were published by ESA in June 1997. The Hipparcos and Tycho data have been used to create the Millennium Star Atlas: an all-sky atlas of one million stars to visual magnitude 11, from the Hipparcos and Tycho Catalogues and 10,000 non-stellar objects included to complement the catalogue data.

There were questions over whether Hipparcos has a systematic error of about 1 milli arc-second in at least some parts of the sky. The value determined by Hipparcos for the distance to the Pleiades is about 10% less than the value obtained by some other methods. By early 2004, the controversy remained unresolved.

Stellarium uses the Hipparcos Catalogue for star data, as well as having traditional names for many of the brighter stars. The stars tab of the search window allows for searching based on a Hipparcos Catalogue number (as well as traditional names), e.g. the star Sadalmelik in the constellation of Aquarius can be found by searching for the name, or it's Hipparcos number, 109074.

The Messier Objects

The Messier objects are a set of astronomical objects catalogued by Charles Messier in his catalogue of Nebulae and Star Clusters first published in 1774. The original motivation behind the catalogue was that Messier was a comet hunter, and was frustrated by objects which resembled but were not comets. He therefore compiled a list of these objects.

The first edition covered 45 objects numbered M1 to M45. The total list consists of 110 objects, ranging from M1 to M110. The final catalogue was published in 1781 and printed in the Connaissance des Temps in 1784. Many of these objects are still known by their Messier number.

Because the Messier list was compiled by astronomers in the Northern Hemisphere, it contains only objects from the north celestial pole to a celestial latitude of about -35°. Many impressive Southern objects, such as the Large and Small Magellanic Clouds are excluded from the list. Because all of the Messier objects are visible with binoculars or small telescopes (under favourable conditions), they are popular viewing objects for amateur astronomers. In early spring, astronomers sometimes gather for "Messier Marathons", when all of the objects can be viewed over a single night.

Stellarium includes images of many Messier objects.

Observing Hints

When star-gazing, there's a few little things which make a lot of difference, and are worth taking into account.

Dark skies For many people getting away from light pollution isn't an easy thing. At best it means a drive away from the towns, and for many the only chance to see a sky without significant glow from street lighting is on vacation. If you can't get away from the cities easily, make the most of it when you are away.

Wrap up warm The best observing conditions are the same conditions that make for cold nights, even in the summer time. Observing is not a strenuous physical activity, so you will feel the cold a lot more than if you were walking around. Wear a lot of warm clothing, don't sit/lie on the floor (at least use a camping mat... consider taking a deck-chair), and take a flask of hot drink.

Dark adaptation The true majesty of the night sky only becomes apparent when the eye has had time to become accustomed to the dark. This process, known as dark adaptation, can take up to half and hour, and as soon as the observer sees a bright light they must start the process over. Red light doesn't compromise dark adaptation as much as white light, so use a red torch if possible (and one that is as dim as you can manage with). A single red LED light is ideal.

The Moon Unless you're particularly interested in observing the Moon on a given night, it can be a nuisance---it can be so bright as to make observation of dimmer objects such as nebulae impossible. When planning what you want to observe, take the phase and position of the Moon into account. Of course Stellarium is the ideal tool for finding this out!

Averted vision A curious fact about the eye is that it is more sensitive to dim light towards the edge of the field of view. If an object is slightly too dim to see directly, looking slightly off to the side but concentrating on the object's location can often reveal it.

Angular distance Learn how to estimate angular distances. Learn the angular distances described in section [sec:handyangles]. If you have a pair of binoculars, find out the angular distance across the field of view and use this as a standard measure.

Handy Angles

Being able to estimate angular distance can be very useful when trying to find objects from star maps in the sky. One way to do this with a device called a crossbow.

Crossbows are a nice way get an idea of angular distances, but carrying one about is a little cumbersome. A more convenient alternative is to hold up an object such as a pencil at arm's length. If you know the length of the pencil, d, and the distance of it from your eye, D, you can calculate it's angular size, θ using this formula:

θ=2•arctan(d/2D)

Another, more handy (ahem!) method is to use the size of your hand at arm's length:

Tip of little finger About 1°

Middle three fingers About 4°

Across the knuckles of the fist About 10°

Open hand About 18°

Using you hand in this way is not very precise, but it's close enough to give you some way to translate an idea like “Mars will be 45° above the Southeastern horizon at 21:30”. Of course, there is variation from person to person, but the variation is compensated for somewhat by the fact that people with long arms tend to have larger hands. In exercise [exercise:handyangles], you will work out your own “handy angles”.